Geodesic Free-Form Curves And Their Properties

In the paper an algorithm to compute geodesic paths on arbitrary discrete surfaces is developed. Based on this key algorithm, three new kinds of curves, called geodesic B-spline curves, geodesic Bézier curves and geodesic Grinding curves are defined respectively, and their important properties and things are studied. From these a new method for modeling on manifold meshes with arbitrary topological structure is put forward. Main contents are as follows:Firstly, Morera's algorithm to compute a geodesic path over a triangulated surface is improved. A discrete geodesic over mesh surfaces with arbitrary topological structure with better approximation is obtained by correcting the iterative process in the algorithm above. Our algorithm only requires finding the intersection point of two straight lines in space, avoids classing mesh vertices and writing equation for a plane through mesh vertices.Secondly, we construct free-from curves over manifold meshes with arbitrary topology, which are the generalization of de Boor, Cutting and Grinding and simple corner-cutting algorithm respectively from Euclidean space to manifold meshes by substituting the line segments connecting the control points with discrete geodesics. The new curve modeling is independent of any parameterization.Thirdly, some properties for the geodesic free-form curves, such as convex hull and convexity preserving properties are derived. Then it is proved that the polygonal lines formed by the simple cutting corners converge to the geodesic free-form curves.Finally, by Visual C++6.0, many examples for computing geodesic curves, then generating geodesic free-form curves on convex or concave triangulations and on quadrangular subdivided surfaces are given ,and these discrete geodesic and geodesic free-form curves are plotted by means of OpenGL tool. The results show that algorithm for discrete geodesic is correct, fast, easy in implementation with computers and better in approximation, and the new curve modeling here has the following advantages: quickness in the rate of convergence for sequences of the corner-cutting polygons, reality in simulation and easiness in shape control as well.